Let's analyze the functions and perform the three operations: addition, subtraction, and multiplication, focusing on the coefficients of the [tex]\(x\)[/tex] terms.
Given:
[tex]\[ f(x) = -3x + 1 \][/tex]
[tex]\[ g(x) = -2x - 3 \][/tex]
### 1. Addition of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f + g)(x) = f(x) + g(x)
\][/tex]
[tex]\[
(f + g)(x) = (-3x + 1) + (-2x - 3)
\][/tex]
Combine like terms:
[tex]\[
(f + g)(x) = -3x - 2x + 1 - 3
\][/tex]
[tex]\[
(f + g)(x) = -5x - 2
\][/tex]
The coefficient of [tex]\(x\)[/tex] in [tex]\(f + g\)[/tex] is [tex]\(-5\)[/tex].
### 2. Subtraction of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f - g)(x) = f(x) - g(x)
\][/tex]
[tex]\[
(f - g)(x) = (-3x + 1) - (-2x - 3)
\][/tex]
Distribute the negative sign:
[tex]\[
(f - g)(x) = -3x + 1 + 2x + 3
\][/tex]
Combine like terms:
[tex]\[
(f - g)(x) = -3x + 2x + 1 + 3
\][/tex]
[tex]\[
(f - g)(x) = -x + 4
\][/tex]
The coefficient of [tex]\(x\)[/tex] in [tex]\(f - g\)[/tex] is [tex]\(-1\)[/tex].
### 3. Multiplication of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
(f \cdot g)(x) = f(x) \cdot g(x)
\][/tex]
[tex]\[
(f \cdot g)(x) = (-3x + 1)(-2x - 3)
\][/tex]
Multiply each term:
[tex]\[
(f \cdot g)(x) = (-3x)(-2x) + (-3x)(-3) + (1)(-2x) + (1)(-3)
\][/tex]
[tex]\[
(f \cdot g)(x) = 6x^2 + 9x - 2x - 3
\][/tex]
Combine like terms:
[tex]\[
(f \cdot g)(x) = 6x^2 + 7x - 3
\][/tex]
The coefficient of [tex]\(x\)[/tex] (the linear term) in [tex]\(f \cdot g\)[/tex] is [tex]\(7\)[/tex].
### Summary of Coefficients
- For [tex]\(f + g\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-5\)[/tex].
- For [tex]\(f - g\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-1\)[/tex].
- For [tex]\(f \cdot g\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(7\)[/tex].
### Conclusion
Among the coefficients [tex]\(-5\)[/tex], [tex]\(-1\)[/tex], and [tex]\(7\)[/tex], the smallest coefficient on the [tex]\(x\)[/tex] term is [tex]\(-5\)[/tex]. Therefore, the operation that results in the smallest coefficient on the [tex]\(x\)[/tex] term is:
[tex]\[ f + g \][/tex]
So, the correct answer is:
[tex]\[ \boxed{f+g} \][/tex]
